Section 1.3 Ordered Pairs and Functions
Total Page：16
File Type：pdf, Size：1020Kb
Load more
Recommended publications
-
Enumeration of Finite Automata 1 Z(A) = 1
INFOI~MATION AND CONTROL 10, 499-508 (1967) Enumeration of Finite Automata 1 FRANK HARARY AND ED PALMER Department of Mathematics, University of Michigan, Ann Arbor, Michigan Harary ( 1960, 1964), in a survey of 27 unsolved problems in graphical enumeration, asked for the number of different finite automata. Re- cently, Harrison (1965) solved this problem, but without considering automata with initial and final states. With the aid of the Power Group Enumeration Theorem (Harary and Palmer, 1965, 1966) the entire problem can be handled routinely. The method involves a confrontation of several different operations on permutation groups. To set the stage, we enumerate ordered pairs of functions with respect to the product of two power groups. Finite automata are then concisely defined as certain ordered pah's of functions. We review the enumeration of automata in the natural setting of the power group, and then extend this result to enumerate automata with initial and terminal states. I. ENUMERATION THEOREM For completeness we require a number of definitions, which are now given. Let A be a permutation group of order m = ]A I and degree d acting on the set X = Ix1, x~, -.. , xa}. The cycle index of A, denoted Z(A), is defined as follows. Let jk(a) be the number of cycles of length k in the disjoint cycle decomposition of any permutation a in A. Let al, a2, ... , aa be variables. Then the cycle index, which is a poly- nomial in the variables a~, is given by d Z(A) = 1_ ~ H ~,~(°~ . (1) ~$ a EA k=l We sometimes write Z(A; al, as, .. -
Formal Construction of a Set Theory in Coq
Saarland University Faculty of Natural Sciences and Technology I Department of Computer Science Masters Thesis Formal Construction of a Set Theory in Coq submitted by Jonas Kaiser on November 23, 2012 Supervisor Prof. Dr. Gert Smolka Advisor Dr. Chad E. Brown Reviewers Prof. Dr. Gert Smolka Dr. Chad E. Brown Eidesstattliche Erklarung¨ Ich erklare¨ hiermit an Eides Statt, dass ich die vorliegende Arbeit selbststandig¨ verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel verwendet habe. Statement in Lieu of an Oath I hereby confirm that I have written this thesis on my own and that I have not used any other media or materials than the ones referred to in this thesis. Einverstandniserkl¨ arung¨ Ich bin damit einverstanden, dass meine (bestandene) Arbeit in beiden Versionen in die Bibliothek der Informatik aufgenommen und damit vero¨ffentlicht wird. Declaration of Consent I agree to make both versions of my thesis (with a passing grade) accessible to the public by having them added to the library of the Computer Science Department. Saarbrucken,¨ (Datum/Date) (Unterschrift/Signature) iii Acknowledgements First of all I would like to express my sincerest gratitude towards my advisor, Chad Brown, who supported me throughout this work. His extensive knowledge and insights opened my eyes to the beauty of axiomatic set theory and foundational mathematics. We spent many hours discussing the minute details of the various constructions and he taught me the importance of mathematical rigour. Equally important was the support of my supervisor, Prof. Smolka, who first introduced me to the topic and was there whenever a question arose. -
Equivalents to the Axiom of Choice and Their Uses A
EQUIVALENTS TO THE AXIOM OF CHOICE AND THEIR USES A Thesis Presented to The Faculty of the Department of Mathematics California State University, Los Angeles In Partial Fulfillment of the Requirements for the Degree Master of Science in Mathematics By James Szufu Yang c 2015 James Szufu Yang ALL RIGHTS RESERVED ii The thesis of James Szufu Yang is approved. Mike Krebs, Ph.D. Kristin Webster, Ph.D. Michael Hoffman, Ph.D., Committee Chair Grant Fraser, Ph.D., Department Chair California State University, Los Angeles June 2015 iii ABSTRACT Equivalents to the Axiom of Choice and Their Uses By James Szufu Yang In set theory, the Axiom of Choice (AC) was formulated in 1904 by Ernst Zermelo. It is an addition to the older Zermelo-Fraenkel (ZF) set theory. We call it Zermelo-Fraenkel set theory with the Axiom of Choice and abbreviate it as ZFC. This paper starts with an introduction to the foundations of ZFC set the- ory, which includes the Zermelo-Fraenkel axioms, partially ordered sets (posets), the Cartesian product, the Axiom of Choice, and their related proofs. It then intro- duces several equivalent forms of the Axiom of Choice and proves that they are all equivalent. In the end, equivalents to the Axiom of Choice are used to prove a few fundamental theorems in set theory, linear analysis, and abstract algebra. This paper is concluded by a brief review of the work in it, followed by a few points of interest for further study in mathematics and/or set theory. iv ACKNOWLEDGMENTS Between the two department requirements to complete a master's degree in mathematics − the comprehensive exams and a thesis, I really wanted to experience doing a research and writing a serious academic paper. -
Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Boise State University, [email protected]
Boise State University ScholarWorks Mathematics Undergraduate Theses Department of Mathematics 5-2014 Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Boise State University, [email protected] Follow this and additional works at: http://scholarworks.boisestate.edu/ math_undergraduate_theses Part of the Set Theory Commons Recommended Citation Rahim, Farighon Abdul, "Axioms of Set Theory and Equivalents of Axiom of Choice" (2014). Mathematics Undergraduate Theses. Paper 1. Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Advisor: Samuel Coskey Boise State University May 2014 1 Introduction Sets are all around us. A bag of potato chips, for instance, is a set containing certain number of individual chip’s that are its elements. University is another example of a set with students as its elements. By elements, we mean members. But sets should not be confused as to what they really are. A daughter of a blacksmith is an element of a set that contains her mother, father, and her siblings. Then this set is an element of a set that contains all the other families that live in the nearby town. So a set itself can be an element of a bigger set. In mathematics, axiom is defined to be a rule or a statement that is accepted to be true regardless of having to prove it. In a sense, axioms are self evident. In set theory, we deal with sets. Each time we state an axiom, we will do so by considering sets. Example of the set containing the blacksmith family might make it seem as if sets are finite. -
Math 311 - Introduction to Proof and Abstract Mathematics Group Assignment # 15 Name: Due: at the End of Class on Tuesday, March 26Th
Math 311 - Introduction to Proof and Abstract Mathematics Group Assignment # 15 Name: Due: At the end of class on Tuesday, March 26th More on Functions: Definition 5.1.7: Let A, B, C, and D be sets. Let f : A B and g : C D be functions. Then f = g if: → → A = C and B = D • For all x A, f(x)= g(x). • ∈ Intuitively speaking, this definition tells us that a function is determined by its underlying correspondence, not its specific formula or rule. Another way to think of this is that a function is determined by the set of points that occur on its graph. 1. Give a specific example of two functions that are defined by different rules (or formulas) but that are equal as functions. 2. Consider the functions f(x)= x and g(x)= √x2. Find: (a) A domain for which these functions are equal. (b) A domain for which these functions are not equal. Definition 5.1.9 Let X be a set. The identity function on X is the function IX : X X defined by, for all x X, → ∈ IX (x)= x. 3. Let f(x)= x cos(2πx). Prove that f(x) is the identity function when X = Z but not when X = R. Definition 5.1.10 Let n Z with n 0, and let a0,a1, ,an R such that an = 0. The function p : R R is a ∈ ≥ ··· ∈ 6 n n−1 → polynomial of degree n with real coefficients a0,a1, ,an if for all x R, p(x)= anx +an−1x + +a1x+a0. -
1 Monoids and Groups
1 Monoids and groups 1.1 Definition. A monoid is a set M together with a map M × M ! M; (x; y) 7! x · y such that (i) (x · y) · z = x · (y · z) 8x; y; z 2 M (associativity); (ii) 9e 2 M such that x · e = e · x = x for all x 2 M (e = the identity element of M). 1.2 Examples. 1) Z with addition of integers (e = 0) 2) Z with multiplication of integers (e = 1) 3) Mn(R) = fthe set of all n × n matrices with coefficients in Rg with ma- trix multiplication (e = I = the identity matrix) 4) U = any set P (U) := fthe set of all subsets of Ug P (U) is a monoid with A · B := A [ B and e = ?. 5) Let U = any set F (U) := fthe set of all functions f : U ! Ug F (U) is a monoid with multiplication given by composition of functions (e = idU = the identity function). 1.3 Definition. A monoid is commutative if x · y = y · x for all x; y 2 M. 1.4 Example. Monoids 1), 2), 4) in 1.2 are commutative; 3), 5) are not. 1 1.5 Note. Associativity implies that for x1; : : : ; xk 2 M the expression x1 · x2 ····· xk has the same value regardless how we place parentheses within it; e.g.: (x1 · x2) · (x3 · x4) = ((x1 · x2) · x3) · x4 = x1 · ((x2 · x3) · x4) etc. 1.6 Note. A monoid has only one identity element: if e; e0 2 M are identity elements then e = e · e0 = e0 1.7 Definition. -
Lecture 3 Partially Ordered Sets
Lecture 3 Partially ordered sets Now for something totally different. In life, useful sets are rarely “just” sets. They often come with additional structure. For example, R isn’t useful just because it’s some set. It’s useful because we can add its elements, multiply its elements, and even compare its elements. If A is, for example, the set of all bananas and apples on earth, it’s not as natural to do any of these things with elements of A. Today, we’ll talk about a structure called a partial order. 3.1 Preliminaries First, let me remind you of something called the Cartesian product, or direct product of sets. Definition 3.1.1. Let A and B be sets. Then the Cartesian product of A and B is defined to be the set consisting of all pairs (a, b) where a A and œ b B. We denote the Cartesian product by œ A B. ◊ Example 3.1.2. R2 is the Cartesian product R R. ◊ Example 3.1.3. Let T be the set of all t-shirts in Hiro’s closet, and S the set of all shorts in Hiro’s closet. Then S T represents the collection of all ◊ possible outfits Hiro is willing to consider in August. (I.e., Hiro is willing to consider putting on any of his shorts together with any of his t-shirts.) 23 24 LECTURE 3. PARTIALLY ORDERED SETS Note that we can “repeat” the Cartesian product operation, or apply it many times at once. Example 3.1.4. -
Lecture 19: Alternating Group
MATH 433 Applied Algebra Lecture 19: Alternating group. Abstract groups. Sign of a permutation Theorem 1 For any n ≥ 2 there exists a unique function sgn : S(n) → {−1, 1} such that • sgn(πσ)= sgn(π) sgn(σ) for all π, σ ∈ S(n), • sgn(τ)= −1 for any transposition τ in S(n). A permutation π is called even if it is a product of an even number of transpositions, and odd if it is a product of an odd number of transpositions. It turns out that π is even if sgn(π)=1 and odd if sgn(π)= −1. Theorem 2 (i) sgn(πσ)= sgn(π) sgn(σ) for any π, σ ∈ S(n). (ii) sgn(π−1)= sgn(π) for any π ∈ S(n). (iii) sgn(id) = 1. (iv) sgn(τ)= −1 for any transposition τ. (v) sgn(σ)=(−1)r−1 for any cycle σ of length r. Alternating group Given an integer n ≥ 2, the alternating group on n symbols, denoted An or A(n), is the set of all even permutations in the symmetric group S(n). Theorem (i) For any two permutations π, σ ∈ A(n), the product πσ is also in A(n). (ii) The identity function id is in A(n). (iii) For any permutation π ∈ A(n), the inverse π−1 is in A(n). In other words, the product of even permutations is even, the identity function is an even permutation, and the inverse of an even permutation is even. Theorem The alternating group A(n) has n!/2 elements. -
Theoretical Probability and Statistics
Theoretical Probability and Statistics Charles J. Geyer Copyright 1998, 1999, 2000, 2001, 2008 by Charles J. Geyer September 5, 2008 Contents 1 Finite Probability Models 1 1.1 Probability Mass Functions . 1 1.2 Events and Measures . 5 1.3 Random Variables and Expectation . 6 A Greek Letters 9 B Sets and Functions 11 B.1 Sets . 11 B.2 Intervals . 12 B.3 Functions . 12 C Summary of Brand-Name Distributions 14 C.1 The Discrete Uniform Distribution . 14 C.2 The Bernoulli Distribution . 14 i Chapter 1 Finite Probability Models The fundamental idea in probability theory is a probability model, also called a probability distribution. Probability models can be specified in several differ- ent ways • probability mass function (PMF), • probability density function (PDF), • distribution function (DF), • probability measure, and • function mapping from one probability model to another. We will meet probability mass functions and probability measures in this chap- ter, and will meet others later. The terms “probability model” and “probability distribution” don’t indicate how the model is specified, just that it is specified somehow. 1.1 Probability Mass Functions A probability mass function (PMF) is a function pr Ω −−−−→ R which satisfies the following conditions: its values are nonnegative pr(ω) ≥ 0, ω ∈ Ω (1.1a) and sum to one X pr(ω) = 1. (1.1b) ω∈Ω The domain Ω of the PMF is called the sample space of the probability model. The sample space is required to be nonempty in order that (1.1b) make sense. 1 CHAPTER 1. FINITE PROBABILITY MODELS 2 In this chapter we require all sample spaces to be finite sets. -
Notes on Sets, Relations, and Functions
Sets, Relations, and Functions S. F. Ellermeyer May 15, 2003 Abstract We give definitions of the concepts of Set, Relation, and Function, andlookatsomeexamples. 1Sets A set is a well—defined collection of objects. An example of a set is the set, A,defined by A = 1, 2, 5, 10 . { } The set A has four members (also called elements). The members of A are the numbers 1, 2, 5, and 10. Another example of a set is the set, B,defined by B = Arkansas, Hawaii, Michigan . { } The set B hasthreemembers—thestatesArkansas,Hawaii,andMichi- gan. In this course, we will restrict our attention to sets whose members are real numbers or ordered pairs of real numbers. An example of a set whose members are ordered pairs of real numbers is the set, C,defined by C = (6, 8) , ( 4, 7) , (5, 1) , (10, 10) . { − − } Note that the set C has four members. When describing a set, we never list any of its members more than once. Thus, the set 1, 2, 5, 5, 10 isthesameastheset 1, 2, 5, 10 . Actually, it is { } { } 1 not even correct to write this set as 1, 2, 5, 5, 10 because, in doing so, we are listing one of the members more than{ once. } If an object, x, is a member of a set, A,thenwewrite x A. ∈ This notation is read as “x is a member of A”, or as “x is an element of A” or as “x belongs to A”. If the object, x, is not a member of the set A,then we write x/A. -
Inverse Monoids Associated with the Complexity Class NP
Inverse monoids associated with the complexity class NP J.C. Birget 6 March 2017 Abstract We study the P versus NP problem through properties of functions and monoids, continuing the work of [3]. Here we consider inverse monoids whose properties and relationships determine whether P is different from NP, or whether injective one-way functions (with respect to worst-case complexity) exist. 1 Introduction We give a few definitions before motivating the monoid approach to the P versus NP problem. Some of these notions appeared already in [3]. By function we mean a partial function A∗ → A∗, where A is a finite alphabet (usually, A = {0, 1}), and A∗ denotes the set of all finite strings over A. Let Dom(f) (⊆ A∗) denote the domain of f, i.e., {x ∈ A∗ : f(x) is defined}; and let Im(f) (⊆ A∗) denote the image (or range) of f, i.e., {f(x) : x ∈ ∗ ∗ Dom(f)}. The length of x ∈ A is denoted by |x|. The restriction of f to X ⊆ A is denoted by f|X , and the identity function on X is denoted by idX or id|X . Definition 1.1 (inverse, co-inverse, mutual inverse) A function f ′ is an inverse of a function f iff f ◦ f ′ ◦ f = f. In that case we also say that f is a co-inverse of f ′. If f ′ is both an inverse and a co-inverse of f, we say that f ′ is a mutual inverse of f, or that f ′ and f are mutual inverses (of each other).1 It is easy to see that f ′ is an inverse of f iff for every y ∈ Im(f): f ′(y) is defined and f ′(y) ∈ f −1(y). -
ALGEBRA II 6.1 Mathematical Functions
Page 1 of 7 ALGEBRA II 6.1 mathematical functions: In mathematics, a function[1] is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x2. The output of a function f corresponding to an input x is denoted by f(x) (read "f of x"). In this example, if the input is −3, then the output is 9, and we may write f(−3) = 9. The input variable(s) are sometimes referred to as the argument(s) of the function. Functions of various kinds are "the central objects of investigation"[2] in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function. In science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation. The input and output of a function can be expressed as an ordered pair, ordered so that the first element is the input (or tuple of inputs, if the function takes more than one input), and the second is the output. In the example above, f(x) = x2, we have the ordered pair (−3, 9).